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Control of Nonlinear Networked Systems With Packet Dropouts: Interval Type-2 Fuzzy Model-Based Approach Hongyi Li, Chengwei Wu, Peng Shi, Senior Member, IEEE, and Yabin Gao
Abstract—In this paper, the problem of fuzzy control for nonlinear networked control systems with packet dropouts and parameter uncertainties is studied based on the interval type-2 fuzzy-model-based approach. In the control design, the intermittent data loss existing in the closed-loop system is taken into account. The parameter uncertainties can be represented and captured effectively via the membership functions described by lower and upper membership functions and relative weighting functions. A novel fuzzy state-feedback controller is designed to guarantee the resulting closed-loop system to be stochastically stable with an optimal performance. Furthermore, to make the controller design more flexible, the designed controller does not need to share membership functions and amount of fuzzy rules with the model. Some simulation results are provided to demonstrate the effectiveness of the proposed results. Index Terms—Fuzzy control, interval type-2 (IT2), nonlinear discrete-time networked control system (NCS), packet dropouts.
I. I NTRODUCTION HE COMPONENTS of practical systems (e.g., the plant, sensor, controller, and actuator) are often distributed at different places and signals are transmitted from one component to another. To facilitate communicating among them, the network is used to constitute the communication links, which produces the networked control systems (NCSs) [1]–[5].
T
Manuscript received June 9, 2014; revised September 7, 2014 and November 7, 2014; accepted November 9, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61333012, Grant 61203002, and Grant 61304003, in part by the Program for New Century Excellent Talents in University under Grant NCET-13-0696, in part by the Chinese National Post-Doctoral Science Foundation under Grant 2014M551111, in part by the Postdoctoral Science Foundation of Northeastern University, in part by the Australian Research Council under Grant DP140102180 and Grant LP140100471, in part by the National Key Basic Research Program, China under Grant 2012CB215202, in part by the 111 Project under Grant B12018, and in part by the National Key Technology R&D Program under Grant 2012BAF19G00. This paper was recommended by Associate Editor S.-F. Su. H. Li is with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). C. Wu and Y. Gao are with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, the University of Adelaide, Adelaide, SA 5005, Australia and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2371814
In view of its obvious superiorities (e.g., low cost, simple installation and maintenance, and high reliability), in recent years, considerable attention has been attracted to study the problems of modeling and control for nonlinear NCSs. Antsaklis and Baillieul [6] pointed out that how to model, analyze and control NCSs has been a topic in the control field under the imperfect communication links. Recently, some results on stability analysis, controller synthesis, and filter design for NCSs were reported in [7]–[10]. Liu et al. [11] presented a new multirate sampled-data output feedback control method for setpoints compensation of complex industrial processes based on a new two-layer networked-based structure. More recently, based on the T-S fuzzy model (FM) approach [12], the problem of controller design for nonlinear NCSs was investigated. It is well know that the T-S FM approach can handle nonlinearities in modeling physical plants by approximating nonlinear terms to any specified accuracy with a family of fuzzy sets and rules [13]–[16]. To mention a few, for the nonlinear NCSs, the stability and stabilization problems were investigated in [9] and [17]–[21], the fault detection problem was studied in [22] and [23], and the filter design problem was considered in [24]–[26], respectively. It is worth noticing that there exist not only nonlinearities but also parameter uncertainties when modeling the plant. The aforementioned results for nonlinear NCSs were obtained via the T-S FM approach based on the type-1 fuzzy set theory. The type-1 FM can capture nonlinearities effectively, however, parameter uncertainties are neglected, which degrades the accuracy of modeling. Since the uncertain parameter exists in the plant, the membership functions become uncertain. Based on the type-2 fuzzy set theory, Mendel [27] proposed an interval type-2 (IT2) fuzzy logic system to handle the nonlinear plant subject to parameter uncertainties. With the lower and upper membership functions, the parameter uncertainties can be handled. Recently, some results concerning the IT2 fuzzy logic system were reported in [28]–[33]. Juang and Hsu [34] proposed a new reinforcement-learning method using online rule generation and Q-value-aided ant colony optimization for fuzzy controller design based on the IT2 FM approach. The authors verified the noise robustness property of IT2 fuzzy systems by comparing with type-1 fuzzy systems. In order to obtain a more general IT2 T-S FM, Lam and Seneviratne [35] extended it. More recently, considerable attention has been paid to the IT2 T-S FM and some theoretical results were
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presented (see [36], [37], and the references therein). It should be pointed out that the IT2 T-S FM and the stability conditions were developed in [35], in which an IT2 T-S FM was provided to describe the T-S fuzzy systems with uncertain membership functions. In [35], it was shown that the IT2 fuzzy statefeedback controller can result in less conservative results than the usual type-1 fuzzy state-feedback one. Furthermore, without matching premise variables, the stability and stabilization conditions were presented in [36] and [37] for IT2 T-S fuzzymodel-based control systems, in which the controller did not need to share the same fuzzy sets and rules with the model, and the flexibility of controller design was enhanced. In [38], a new filter design approach was proposed for IT2 fuzzy systems with D stability constraints in a unified frame. It is well known that the parameter uncertainties also exist in nonlinear NCSs. However, it should be pointed out that there exist few IT2 controller design results on nonlinear NCSs. Therefore, it is challenging and practical to study the controller design problem for nonlinear NCSs with parameter uncertainties in the framework of the IT2 T-S FM. This paper focuses on designing the IT2 state-feedback controller for the nonlinear NCSs with random data missing. The main contributions of this paper can be summarized as follows. 1) The parameter uncertainty of the plant can be handled by using membership functions with lower and upper bounds. 2) The membership functions and number of fuzzy rules of the controller to be designed are independent of those of the plant, which makes the controller design more flexible, and results in less conservativeness. 3) The state-feedback controller is first time designed for nonlinear NCSs with parameter uncertainties on the basis of the IT2 T-S FM. The remainder of this paper is organized as follows. The considered problem is described in Section II. The main results are presented in Section III. A numerical example is utilized to verify the usefulness of the proposed method in Section IV. Finally, Section V concludes this paper. A. Notation Notations utilized in this paper are quite standard. The superscript “T” denotes matrix transposition and the identity matrix and zero matrix with compatible dimensions are represented by I and 0, respectively. Rn devotes the n-dimensional Euclidean space. The notation P > 0(≥ 0) suggests that P is positive definite (semi-definite) with the real symmetric structure. The notation A indicates the norm of a matrix A and and is defined by A = tr(AT A). l2 [0, ∞) means the space of square-integrable vector functions over [0, ∞); Rn describes the n-dimensional Euclidean space, Rm×n indicates the set of all real matrices of dimension m × n, and . 2 shows the usual l2 [0, ∞) norm. In complex matrix, the symbol (∗) is utilized to represent a symmetric term, and we utilize diag {. . .} to describe the matrix with block diagonal structure. Furthermore, Prob{·} represents the occurrence probability of the event “·”; E{x|y} and E{x} signifies expectation of x conditional on y and expectation of x, respectively. Matrices in this
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paper without dimensions explicitly stated, we assume they have compatible dimensions. II. P ROBLEM F ORMULATION In this section, we consider the following nonlinear NCSs with data packet dropouts in the framework of the IT2 T-S FM and a r-rule FM is as follows. A. IT2 Plant Model Plant Rule i: IF f1 (σ (k)) is Mi1 , and f2 (σ (k)) is Mi2 and, . . . , and fθ (σ (k)) is Miθ , THEN x (k + 1) = Ai x (k) + Bi u (k) + Ei w (k) z (k) = Ci x (k) + Di u (k) + Fi w (k) , i = 1, 2, . . . , r (1) where Mij stands for the fuzzy set, and f (σ (k)) = [ f1 (σ (k)), f2 (σ (k)), . . . , fθ (σ (k))] denotes the premise variable and σ (k) may depend on the system state. x(k) ∈ Rn stands for the state; z(k) ∈ Rnz stands for the controlled output; u(k) ∈ Rnu stands for the control input and w(t) ∈ Rnw stands for the disturbance input which belongs to l2 [0, ∞). Ai , Bi , Ei , Ci , Di , and Fi are system matrices with appropriate dimensions. The scalar r is the number of IF-THEN rules of the system. The interval sets listed below present the firing strength of the ith rule Wi (σ (k)) = mi (σ (k)), mi (σ (k)) , i = 1, 2, . . . , r where mi (σ (k)) =
θ
uMip fp (σ (k)) ≥ 0
p=1
mi (σ (k)) =
θ
uMip fp (σ (k)) ≥ 0
p=1
uMip fp (σ (k)) ≥ uMip fp (σ (k)) ≥ 0
mi (σ (k)) ≥ mi (σ (k)) ≥ 0
(2)
uMip ( fp (σ (k))), uMip ( fp (σ (k))), mi (σ (k)), and mi (σ (k)) denote lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively. The inferred dynamics of (1) is as follows: x (k + 1) = z (k) =
r i=1 r
mi (σ (k)) [Ai x (k) + Bi u (k) + Ei w (k)] mi (σ (k)) [C1i x (k) + Di u (k) + Fi w (k)] (3)
i=1
where mi (σ (k)) = ai (σ (k))mi (σ (k)) + ai (σ (k))mi (σ (k)) (4) r
mi (σ (k)) ≥ 0, 0 ≤ ai (σ (k)) ≤ 1, 0 ≤ ai (σ (k)) ≤ 1 mi (σ (k)) = 1, ai (σ (k)) + ai (σ (k)) = 1
(5)
i=1
with ai (σ (k)) and ai (σ (k)) being nonlinear weighting functions and mi (σ (k)) regarded as the grades of membership.
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Remark 1: The parameter uncertainties can result in the uncertainties of the membership functions of the plant when we model the plant. In IT2 T-S FM, the uncertain parameter is used to determine the lower and upper membership functions. With the lower and upper membership functions and relevant weighting functions, we can determine the desired membership functions according to the forms of (4) and (8). Thus, the parameter uncertainties can be captured, that is, the uncertain parameter exist in the model in the form of lower and upper bounds. Lam and Seneviratne [35] used an example to demonstrate it specifically. Remark 2: The weighting functions ai (σ (k)) and ai (σ (k)) play an important role in determining the desired membership functions. As the parameter exists in the plant in the form of the uncertainty, the weighting functions should change when the uncertain parameter changes within its range. Thus, they are not necessary to be known. Liang and Mendel [32] chose them as 0.5. In this paper, both are defined as nonlinear functions, which are related to the system state so that they can describe the dynamics of membership functions well. In this paper, the IT2 state-feedback controller does not share membership functions and amount of fuzzy rules with the model for enhancing the flexibility of the controller design. The details of a s-rule controller are as follows. B. IT2 Controller Design Controller Rule j: IF g1 (σ (k)) is Nj1 , and g2 (σ (k)) is Nj2 and, . . . , and g¯ (σ (k)) is Nj¯ , THEN uc (k) = Kj xc (k) , j = 1, 2, . . . , s
(6)
where xc (k) ∈ Rn is the controller state; uc (k) ∈ Rnu is the output of the controller and Kj are controller gains to be determined; s is the number of rules of the controller. The following interval sets describe the firing strength of the jth rule:
j (σ (k)) = ωj (σ (k)), ωj (σ (k)) where ωi (σ (k)) =
¯
uNjq gq (σ (k)) ≥ 0
Fig. 1.
3
Plant prototype.
where ω˘ j (σ (k)) ωj (σ (k)) = s ˘ j (σ (k)) j=1 ω
(8)
ω˘ j (σ (k)) = bj (σ (k))ωj (σ (k)) + bj (σ (k))ωj (σ (k)) 0 ≤ bj (σ (k)) ≤ 1, ωj (σ (k)) ≥ 0, 0 ≤ bj (σ (k)) ≤ 1 s ωj (σ (k)) = 1, bj (σ (k)) + bj (σ (k)) = 1 (9) j=1
bi (σ (k)) and bj (σ (k)) are predefined weighting functions and ωj (σ (k)) is regarded as the grades of membership. Remark 3: In the IT2 controller, the weighting functions bi (σ (k)) and bj (σ (k)) are predefined. Lam et al. [37] have used different values of weighting functions to illustrate whether they have effect on the stability capacity and it has been proven that reasonable values can reduce the conservativeness. To this end, with the above constraints of them, they are also defined as two nonlinear functions related to the system state. Remark 4: For the purpose of reducing the conservativeness, the forms of (4) and (8) are different. If (8) is the same as that of (4), it is obvious that the same form is sjust a specific case of (8) we have designed (when j=1 (bj (σ (k))ωj (σ (k)) + bj (σ (k))ωj (σ (k))) = 1). Therefore, (8) is more general and helpful to enhance the flexibility of controller design.
q=1
ωi (σ (k)) =
¯
C. Communication Links
uNjq gq (σ (k)) ≥ 0
q=1
uNjq gq (σ (k)) ≥ uNjq gq (σ (k)) ≥ 0 ωi (σ (k)) ≥ ωi (σ (k)) ≥ 0 uNjq (gp (σ (k))), uNjq (gp (σ (k))), ωi (σ (k)), and ωi (σ (k)) denote lower membership functions, upper membership functions, the lower grade of membership, and the upper grade of membership, respectively. Then, the overall fuzzy controller is as follows: uc (k) =
s j=1
ωj (σ (k)) Kj xc (k)
(7)
The plant prototype with communication network is shown in Fig. 1. The intermittent data missing phenomenon introduced by network should be considered in this paper. The states of the plant and the controller are not equal [e.g., x(k) = xc (k)], and the output of the controller is not equal to the input of the plant [e.g., uc (k) = u(k)]. A stochastic approach is employed to describe the data missing phenomenon and it is modeled as xc (k) = α (k) x (k) , u (k) = β (k) uc (k) with α(k) and β(k) obeying the Bernoulli process. The processes α(k) and β(k), respectively, describe the imperfect communication link between the sensor and the controller and
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between the controller and the actuator, which are assumed as follows: ¯ α (k) = 1 E {α (k)} = α, Prob {α (k)} = 1 − α¯ α (k) = 0 ¯ β (k) = 1 E {β (k)} = β, . Prob {β (k)} = ¯ 1 − β, β (k) = 0 We introduce the variable e(k) α(k)β(k), then we can get E {e (k)} = e¯ , e (k) = 1 Prob {e (k)} = . 1 − e¯ , e (k) = 0
where ϑ iji1 i2 ...in zl and ϑ iji1 i2 ...in zl are constant scalars to be computed; 0 ≤ υτ is zl (στ (k)) ≤ 1 with = 1 for the property υτ 1zl (στ (k)) + υτ 2zl (στ (k)) τ, s = 1, 2, . . . , n; l = 1, 2, . . . , ς + 1; iτ = 1, 2; ; and υτ σ (k) ∈ Hz is zl (στ (k))
n = 0if otherwise. Thus, q 2 2 2 . . . i1 =1 i2 =1 in =1 τ =1 υτ iτ zl (στ (k)) = 1 for z=1 all l is achieved, which is used to analyze the system stability. Then, (12) can be rewritten as follows: x (k + 1) =
According to the assumption, we have u (k) =
s
ωj (σ (k)) e (k) Kj (k) x (k).
z (k) =
s
i=1
hij (σ (k)) = mi (σ (k)) ωj (σ (k)) =
ωj (σ (k))
j=1 r s
=
mi (σ (k)) ωj (σ (k)) = 1.
z (k) =
i=1 j=1 r s
(11)
mi (σ (k)) ωj (σ (k)) Aij x (k) + Ei w (k) mi (σ (k)) ωj (σ (k)) Cij x (k) + Fi w (k) (12)
where Aij = Ai + e¯ Bi Kj + e˜ (k)Bi Kj , e˜ (k) = e(k) − e¯ Cij = Ci + e¯ Di Kj + e˜ (k)Di Kj .
E {˜e(k)} =
e¯ (1 − e¯ ) , e˜ (k) = e˜ (k)˜e(k) 0, e˜ (k) = e˜ (k).
The following work is to design the controller for the IT2 T-S fuzzy control system (12). The state space H consists of q connected sub-state spaces Hz (z = 1, 2, . . . , q) q with H = ∪z=1 Hz . Then we divide the footprint of uncertainty (FOU) into ς + 1 sub-FOUs. For l = 1, 2, . . . , ς + 1, the expressions of lower and upper membership functions in the lth sub-FOU are as follows: hijl (σ (k)) = hijl (σ (k)) =
q 2 2 z=1 i1 =1 i2 =1 q 2 2 z=1 i1 =1 i2 =1
...
2 n in =1 τ =1
...
n 2 in =1 τ =1
r s
ρijl (σ (k)) (16)
hij (σ (k)) = 1, 0 ≤ ζ ijl (σ (k)) ≤ ζ ijl (σ (k)) ≤ 1 (17)
i=1 j=1
i=1 j=1
It is obvious that
ς+1
× (ζ ijl (σ (k))hijl (σ (k)) + ζ ijl (σ (k))hijl (σ (k)))
According to (3) and (10), the closed-loop system is represented as r s
(15)
l=1
i=1 j=1
x (k + 1) =
hij (σ (k)) Cij x (k) + Fi w (k)
where
From (5) and (9), we know mi (σ (k)) =
i=1 j=1 r s
hij (σ (k)) Aij x (k) + Ei w (k)
i=1 j=1
(10)
j=1
r
r s
ρijl (σ (k)) =
1, 0,
hij (σ (k)) ∈ sub-FOU l else
(18)
where ζ ijl (σ (k)) and ζ ijl (σ (k)) are two functions and unnec-
essary to be known. ζ ijl (σ (k))+ζ ijl (σ (k)) = 1 for i, j and l.
For brevity, hij hij (σ (k)). Before proceeding further, we introduce the following definition. Definition 1 [17]: Under the initial condition x(0) and w(k) ≡ 0, (15) is stochastically stable if there is a matrix W > 0 such that ∞ |x(k)|2 |x(0) < x(0)T Wx(0). E k=0
Our aim in this paper is to design a fuzzy state-feedback controller for nonlinear NCSs with imperfect communication links in the framework of the IT2 T-S FM to satisfy the following two requirements. 1) Equation (15) is stochastically stable. 2) Under zero initial condition, with a given positive scalar γ , the controlled output z(k) satisfies ⎧ ⎫ ∞ ⎨ ⎬ |z (k)|2 ≤ γ w2 . E ⎩ ⎭ k=0
III. M AIN R ESULTS υτ iτ zl (στ (k))ϑ iji1 i2 ...in zl (13) υτ iτ zl (στ (k))ϑ iji1 i2 ...in zl (14)
0 ≤ hijl (σ (k)) ≤ hijl (σ (k)) ≤ 1 ϑ iji1 i2 ...in zl ≤ ϑ iji1 i2 ...in zl
In this section, for given controller gain matrices Kj ( j = 1, 2, . . . , s), we first provide the following theorem, which ensures (15) is stochastically stable with a given H∞ performance. Theorem 1: With the FOU and state space divided into ς + 1 sub-FOUs and q connected sub-state spaces, for the given controller gain matrices Kj ( j = 1, 2, . . . , s) and positive scalar γ , (15) is stochastically stable and satisfies
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a given H∞ performance, if there exist symmetric matrices P > 0, Nijl > 0, Yijl > 0, Xijl > 0, Vijl > 0 (i = 1, 2, . . . , r, j = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1), and M with appropriate dimensions such that the following inequalities hold: r s ¯ 1ijl − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl Wijl ϑ iji1 i2 ...in zl Q 1 2 n i=1 j=1
r s ς+1
ρijl (σ (k)) ζ ijl (σ (k)) + ζ ijl (σ (k))
i=1 j=1 l=1
× hijl (σ (k)) Nijl ≥ 0 −
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))(hijl (σ (k))
i=1 j=1 l=1
(19)
¯ 2ijl + Wijl + M > 0 Q
(20)
−
r s ς+1
¯ 1ijl = ˜ Tij Pˇ ¯ Tij E˜ ijl ˜ ij + ¯ ij − P, ˆ Pˆ = diag P, γ 2 I Q ¯ 2ijl = ˜ ij + ¯ ij − P, ˜ Tij Pˇ 1 ¯ Tij E˜ 1ijl ˆ f = e¯ (1 − e¯ ) Q ¯ Ei ˜ ij = Aij , Pˇ = diag P + Nijl , P + Nijl fBi Kj 0 ¯ Fi Cij ¯ ij = , E˜ ijl = diag I + Xijl , I + Xijl fDi Kj 0 ˇP1 = diag P − Yijl , P − Yijl E˜ 1ijl = diag I − Vijl , I − Vijl A¯ ij = Ai + eBi Kj , C¯ ij = Ci + eDi Kj .
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))(hijl (σ (k)) − hijl (σ (k)) Vijl ≥ 0
(26)
ρijl (σ (k))ζ ijl (σ (k))hijl (σ (k)) Xijl ≥ 0.
(27)
i=1 j=1 l=1
Adding (22)–(25) into (21), it can be found that V (k) ≤ x¯ T (k)
where P > 0 is the matrix to achieve a feasible solution. Let x¯ T (k) = xT (k) wT (k) . Then, one can have the following: V (k) = E {V (k + 1) |¯x (k)} − V (k) T r s Aij PAij ATij PEi ≤ x¯ T (k) x¯ (k) hij ∗ EiT PEi i=1 j=1 P 0 T − x¯ (k) x¯ (k) 0 0 r s ¯ ij − P A¯ Tij PEi ijT P T x¯ (k) = x¯ (k) hij ∗ EiT PEi i=1 j=1
(21)
where
Q1ijl =
ijl
Q2ijl
When w(k) = 0, it can be seen from (19) and (20) that < 0, where =
r s ς+1
r s ς+1
(22)
ρijl (σ (k)) 1 − ζ ijl (σ (k))
˘ 1ij − (hijl (σ (k)) hijl (σ (k)) Q
− hijl (σ (k)))W1ijl + hijl (σ (k)) M1 − M1 + ζ ijl (σ (k)) hijl (σ (k)) − hijl (σ (k)) ˘ 2ij + W1ijl + M1 (29) × Q
W1ijl Wijl = ∗
W2ijl M1 , M= W3ijl ∗
M2 . M3
Then, we can get E xT (k + 1) Px (k + 1) − xT (k) Px (k)
i=1 j=1 l=1
× hijl (σ (k)) − hijl (σ (k)) Wijl ≥ 0
ρijl (σ (k))
i=1 j=1 l=1
in which
#
+ 1 − ζ ijl (σ (k)) × hijl (σ (k)) − 1 M = 0 −
˘ 2ij A¯ T (P − Yijl )Ei Q ij ˘ 2ij = ijT Pˇ 1 ij − P. ,Q = ∗ EiT (P − Yijl )Ei
×
A¯ ij , P¯ = diag{P, P}. fBi Kj
˘ 1ij A¯ T (P + Nijl )Ei Q ij ˘ 1ij = ijT P ˇ ij − P ,Q ∗ EiT (P + Nijl )Ei
We introduce some slack matrices ⎡ r s ς+1 ⎣ ρijl (σ (k)) ζ (σ (k))hijl (σ (k))
ρijl (σ (k))
i=1 j=1 l=1
V(k) = xT (k) Px (k)
i=1 j=1 l=1
r s ς+1
× hijl (σ (k)) × Q1ijl − hijl σ (k) − hijl (σ (k))
× Wijl + hijl (σ (k)) M − M + ζ ijl (σ (k)) hijl (σ (k)) − hijl (σ (k)) (28) × Q2ijl + Wijl + M x¯ (k)
Proof: We firstly prove the stability of (15). Consider the following Lyapunov function for (15):
(25)
i=1 j=1 l=1
where
ij =
(24)
− hijl (σ (k)) Yijl ≥ 0
+ ϑ iji1 i2 ...in zl M − M < 0
where
5
(23)
≤ − λmin (−) xT (k) x (k).
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Computing the sum of the mathematical expectation for both sides of the inequality from k = 0, 1, . . . , d with any d ≥ 1, one can obtain
E xT (d + 1) Px (d + 1) − xT (0) Px (0) d 2 |x (k)| ≤ −λmin (−) E
Adding (26) and (27) to (30), we get J ≤ xT (k)
r s ς+1
+ V(k) +
E
d
|x (k)|
2
$
T
ij ij + ζ ijl (σ (k)))hijl (σ (k))) ∗
k=0
which yields
ρijl (σ (k))(ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
r s ς+1
# C¯ ijT Fi x (k) FiT Fi − γ 2 I
ρijl (σ (k))ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
≤ (λmin (−))−1 xT (0) Px (0)
k=0
− E xT (d + 1) Px (d + 1) ≤ (λmin (−))−1 xT (0) Px (0).
When d = 1, . . . , ∞, considering E{xT (∞)Px(∞)} ≥ 0, we have d 2 |x (k)| ≤ (λmin (−))−1 xT (0) Px (0) E
× zT (k) Xijl z (k) −
ρijl (σ (k))
i=1 j=1 l=1
× ζ ijl (σ (k))(hijl (σ (k)) − hijl (σ (k)))zT (k) Vijl z (k) = x¯ T (k)
r s ς+1
ρijl (σ (k))
i=1 j=1 l=1
×
¯ 1ijl − (hijl (σ (k)) − hijl (σ (k)))Wijl hijl (σ (k)) Q + hijl (σ (k)) M − M ϑ iji1 i2 ...in zl − ϑ iji1 i2 ...in zl x¯ (k)
k=0
= xT (0) Wx (0)
r s ς+1
+ x¯ T (k)
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))
i=1 j=1 l=1
¯ 2ijl + Wijl + M)¯x (k). × (hijl (σ (k)) − hijl (σ (k)))(Q
where W (λmin (−))−1 P
Substituting (13) and (14) into (31), we have
which means W > 0. From Definition 1, the closed-loop fuzzy system in (15) is stochastically stable. The H∞ performance of (15) will be considered. Under the zero initial condition, the H∞ performance index is J E {V (k + 1) |¯x (k)} − V (k) +E zT (k) z (k) |¯x (k) − γ 2 wT (k) w (k) . Then, we get J = E zT (k) z (k) |¯x (k) − γ 2 wT (k) w (k) + V (k) T r s TF Cij Cij Cijl i T hij x (k) ≤ x (k) ∗ FiT Fi i=1 j=1 0 0 − xT (k) x (k) + V(k) 0 γ 2I = xT (k)
+1 r s ς
ρijl (σ (k))(ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
+ ζ ijl (σ (k)))hijl (σ (k)))
¯ ijT ¯ ij ∗
+ V(k) where
T ¯ ij = C¯ ijT fKjT DTi and V(k) is as in (28).
(31)
C¯ ijT Fi x (k) FiT Fi − γ 2 I (30)
J ≤ x¯ T (k)
r s ς+1 i=1 j=1 l=1
×
2 i2 =1
×
...
2 n
ρijl (σ (k))
q 2 z=1 i1 =1
υτ iτ zl (στ (k))
in =1 τ =1
¯ 1ijl − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl Wijl ϑ iji1 i2 ...in zl Q 1 2 n
+ ϑ iji1 i2 ...in zl M − M + ζ ijl (σ (k)) ϑ iji1 i2 ...in zl − ϑ iji1 i2 ...in zl ¯ 2ijl + Wijl + M ϑ iji i ...i zl × Q 1 2 n −ϑ iji1 i2 ...in zl x¯ (k). (32)
to the proof in [37], at any time, the equality Referring ς+1 ρ (σ (k)) = 1 is ensured by only one ρijl (σ (k)) = 1 ijl l=1 for each fixed i, j value. According to (19) and (20), one can have E zT (k) z (k) − γ 2 wT (k) w (k) + V (k) ≤ 0 % ∞ 2 which yields J ≤ 0. Then, we can obtain E{ k=0 |z(k)| } ≤ γ w2 . The proof is completed. Based on Theorem 1, the existence conditions of the controller can be developed in the following theorem. Theorem 2: Considering the fuzzy system in (15), for a positive scalar γ , (15) is stochastically stable and satisfies a given H∞ performance, if there exist symmetric matrices L > 0, ¯ 1ijl > 0, W ¯ 2ijl > 0, W ¯ 3ijl > 0, N¯ ijl > 0, Xijl > 0, Y¯ ijl > 0, W
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¯ 1, M ¯ 2, M ¯ 3 satisfying the following Vijl > 0 and matrices K¯ j , M conditions: ¯ ijl ˘ ijl 0 (34) ∗ P˘ L − Y¯ ijl < 0 I − Vijl < 0
(35) (36)
where
P˜ ijl = diag N¯ ijl − L, N¯ ijl − L, Xijl − I, Xijl − I P˘ = diag 3L − Y¯ ijl , 3L − Y¯ ijl , 3I − Vijl , 3I − Vijl % ¯ 2ijl ¯ 1ijl ¯ = ϑ iji1 i2 ...in zl , ijl = ¯ 3ijl ∗
˘ ijl =
1ijl EiT
f K¯ jT BTi 2ijl f K¯ jT DTi 0 FiT 0 T T T T ¯ ¯ f Kj Bi 4ijl f Kj Di
˘ ijl = 3ijl ET 0 FiT 0 i ¯ ¯2 ¯ ¯ W2ijl + M −L + W1ijl + M1 Uijl = ¯ 3ijl + M ¯3 ∗ −γ 2 I + W 1ijl = LATi + e¯ K¯ jT BTi , 2ijl = LCiT + e¯ K¯ jT DTi 3ijl = LATi + e¯ K¯ jT BTi , 4ijl = LCiT + e¯ K¯ jT DTi ¯ 1ijl = −ϑ iji1 i2 ...in zl L − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 1ijl 1 2 n ' & 1 ¯1 + ϑ iji1 i2 ...in zl − M rs ¯ 2ijl = − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 2ijl 1 2 n ' & 1 ¯2 + ϑ iji1 i2 ...in zl − M rs ¯ 3ijl = −ϑ iji1 i2 ...in zl γ 2 I − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 3ijl 1 2 n ' & 1 ¯ 3. + ϑ iji1 i2 ...in zl − M rs Furthermore, if the aforementioned conditions hold, the controller gain matrices in (6) can be designed as follows: Kj = K¯ j L−1 . Proof: Firstly, some matrix variables are defined as follows: ¯ 1ijl = LW1ijl L, W ¯ 2ijl = LW2ijl , W ¯ 3ijl = W3ijl L = P−1 , W ¯ 1 = LM1 L, M ¯ 2 = LM2 , M ¯ 3 = M3 K¯ j = Kj L, M N¯ ijl = LNijl L, Y¯ ijl = LYijl L. Pre- and post-multiplying (33) by diag{P1 , P2 } and its transposition, we can obtain ijl ijl
1
Control of Nonlinear Networked Systems With Packet Dropouts: Interval Type-2 Fuzzy Model-Based Approach Hongyi Li, Chengwei Wu, Peng Shi, Senior Member, IEEE, and Yabin Gao
Abstract—In this paper, the problem of fuzzy control for nonlinear networked control systems with packet dropouts and parameter uncertainties is studied based on the interval type-2 fuzzy-model-based approach. In the control design, the intermittent data loss existing in the closed-loop system is taken into account. The parameter uncertainties can be represented and captured effectively via the membership functions described by lower and upper membership functions and relative weighting functions. A novel fuzzy state-feedback controller is designed to guarantee the resulting closed-loop system to be stochastically stable with an optimal performance. Furthermore, to make the controller design more flexible, the designed controller does not need to share membership functions and amount of fuzzy rules with the model. Some simulation results are provided to demonstrate the effectiveness of the proposed results. Index Terms—Fuzzy control, interval type-2 (IT2), nonlinear discrete-time networked control system (NCS), packet dropouts.
I. I NTRODUCTION HE COMPONENTS of practical systems (e.g., the plant, sensor, controller, and actuator) are often distributed at different places and signals are transmitted from one component to another. To facilitate communicating among them, the network is used to constitute the communication links, which produces the networked control systems (NCSs) [1]–[5].
T
Manuscript received June 9, 2014; revised September 7, 2014 and November 7, 2014; accepted November 9, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61333012, Grant 61203002, and Grant 61304003, in part by the Program for New Century Excellent Talents in University under Grant NCET-13-0696, in part by the Chinese National Post-Doctoral Science Foundation under Grant 2014M551111, in part by the Postdoctoral Science Foundation of Northeastern University, in part by the Australian Research Council under Grant DP140102180 and Grant LP140100471, in part by the National Key Basic Research Program, China under Grant 2012CB215202, in part by the 111 Project under Grant B12018, and in part by the National Key Technology R&D Program under Grant 2012BAF19G00. This paper was recommended by Associate Editor S.-F. Su. H. Li is with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). C. Wu and Y. Gao are with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, the University of Adelaide, Adelaide, SA 5005, Australia and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2371814
In view of its obvious superiorities (e.g., low cost, simple installation and maintenance, and high reliability), in recent years, considerable attention has been attracted to study the problems of modeling and control for nonlinear NCSs. Antsaklis and Baillieul [6] pointed out that how to model, analyze and control NCSs has been a topic in the control field under the imperfect communication links. Recently, some results on stability analysis, controller synthesis, and filter design for NCSs were reported in [7]–[10]. Liu et al. [11] presented a new multirate sampled-data output feedback control method for setpoints compensation of complex industrial processes based on a new two-layer networked-based structure. More recently, based on the T-S fuzzy model (FM) approach [12], the problem of controller design for nonlinear NCSs was investigated. It is well know that the T-S FM approach can handle nonlinearities in modeling physical plants by approximating nonlinear terms to any specified accuracy with a family of fuzzy sets and rules [13]–[16]. To mention a few, for the nonlinear NCSs, the stability and stabilization problems were investigated in [9] and [17]–[21], the fault detection problem was studied in [22] and [23], and the filter design problem was considered in [24]–[26], respectively. It is worth noticing that there exist not only nonlinearities but also parameter uncertainties when modeling the plant. The aforementioned results for nonlinear NCSs were obtained via the T-S FM approach based on the type-1 fuzzy set theory. The type-1 FM can capture nonlinearities effectively, however, parameter uncertainties are neglected, which degrades the accuracy of modeling. Since the uncertain parameter exists in the plant, the membership functions become uncertain. Based on the type-2 fuzzy set theory, Mendel [27] proposed an interval type-2 (IT2) fuzzy logic system to handle the nonlinear plant subject to parameter uncertainties. With the lower and upper membership functions, the parameter uncertainties can be handled. Recently, some results concerning the IT2 fuzzy logic system were reported in [28]–[33]. Juang and Hsu [34] proposed a new reinforcement-learning method using online rule generation and Q-value-aided ant colony optimization for fuzzy controller design based on the IT2 FM approach. The authors verified the noise robustness property of IT2 fuzzy systems by comparing with type-1 fuzzy systems. In order to obtain a more general IT2 T-S FM, Lam and Seneviratne [35] extended it. More recently, considerable attention has been paid to the IT2 T-S FM and some theoretical results were
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presented (see [36], [37], and the references therein). It should be pointed out that the IT2 T-S FM and the stability conditions were developed in [35], in which an IT2 T-S FM was provided to describe the T-S fuzzy systems with uncertain membership functions. In [35], it was shown that the IT2 fuzzy statefeedback controller can result in less conservative results than the usual type-1 fuzzy state-feedback one. Furthermore, without matching premise variables, the stability and stabilization conditions were presented in [36] and [37] for IT2 T-S fuzzymodel-based control systems, in which the controller did not need to share the same fuzzy sets and rules with the model, and the flexibility of controller design was enhanced. In [38], a new filter design approach was proposed for IT2 fuzzy systems with D stability constraints in a unified frame. It is well known that the parameter uncertainties also exist in nonlinear NCSs. However, it should be pointed out that there exist few IT2 controller design results on nonlinear NCSs. Therefore, it is challenging and practical to study the controller design problem for nonlinear NCSs with parameter uncertainties in the framework of the IT2 T-S FM. This paper focuses on designing the IT2 state-feedback controller for the nonlinear NCSs with random data missing. The main contributions of this paper can be summarized as follows. 1) The parameter uncertainty of the plant can be handled by using membership functions with lower and upper bounds. 2) The membership functions and number of fuzzy rules of the controller to be designed are independent of those of the plant, which makes the controller design more flexible, and results in less conservativeness. 3) The state-feedback controller is first time designed for nonlinear NCSs with parameter uncertainties on the basis of the IT2 T-S FM. The remainder of this paper is organized as follows. The considered problem is described in Section II. The main results are presented in Section III. A numerical example is utilized to verify the usefulness of the proposed method in Section IV. Finally, Section V concludes this paper. A. Notation Notations utilized in this paper are quite standard. The superscript “T” denotes matrix transposition and the identity matrix and zero matrix with compatible dimensions are represented by I and 0, respectively. Rn devotes the n-dimensional Euclidean space. The notation P > 0(≥ 0) suggests that P is positive definite (semi-definite) with the real symmetric structure. The notation A indicates the norm of a matrix A and and is defined by A = tr(AT A). l2 [0, ∞) means the space of square-integrable vector functions over [0, ∞); Rn describes the n-dimensional Euclidean space, Rm×n indicates the set of all real matrices of dimension m × n, and . 2 shows the usual l2 [0, ∞) norm. In complex matrix, the symbol (∗) is utilized to represent a symmetric term, and we utilize diag {. . .} to describe the matrix with block diagonal structure. Furthermore, Prob{·} represents the occurrence probability of the event “·”; E{x|y} and E{x} signifies expectation of x conditional on y and expectation of x, respectively. Matrices in this
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paper without dimensions explicitly stated, we assume they have compatible dimensions. II. P ROBLEM F ORMULATION In this section, we consider the following nonlinear NCSs with data packet dropouts in the framework of the IT2 T-S FM and a r-rule FM is as follows. A. IT2 Plant Model Plant Rule i: IF f1 (σ (k)) is Mi1 , and f2 (σ (k)) is Mi2 and, . . . , and fθ (σ (k)) is Miθ , THEN x (k + 1) = Ai x (k) + Bi u (k) + Ei w (k) z (k) = Ci x (k) + Di u (k) + Fi w (k) , i = 1, 2, . . . , r (1) where Mij stands for the fuzzy set, and f (σ (k)) = [ f1 (σ (k)), f2 (σ (k)), . . . , fθ (σ (k))] denotes the premise variable and σ (k) may depend on the system state. x(k) ∈ Rn stands for the state; z(k) ∈ Rnz stands for the controlled output; u(k) ∈ Rnu stands for the control input and w(t) ∈ Rnw stands for the disturbance input which belongs to l2 [0, ∞). Ai , Bi , Ei , Ci , Di , and Fi are system matrices with appropriate dimensions. The scalar r is the number of IF-THEN rules of the system. The interval sets listed below present the firing strength of the ith rule Wi (σ (k)) = mi (σ (k)), mi (σ (k)) , i = 1, 2, . . . , r where mi (σ (k)) =
θ
uMip fp (σ (k)) ≥ 0
p=1
mi (σ (k)) =
θ
uMip fp (σ (k)) ≥ 0
p=1
uMip fp (σ (k)) ≥ uMip fp (σ (k)) ≥ 0
mi (σ (k)) ≥ mi (σ (k)) ≥ 0
(2)
uMip ( fp (σ (k))), uMip ( fp (σ (k))), mi (σ (k)), and mi (σ (k)) denote lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively. The inferred dynamics of (1) is as follows: x (k + 1) = z (k) =
r i=1 r
mi (σ (k)) [Ai x (k) + Bi u (k) + Ei w (k)] mi (σ (k)) [C1i x (k) + Di u (k) + Fi w (k)] (3)
i=1
where mi (σ (k)) = ai (σ (k))mi (σ (k)) + ai (σ (k))mi (σ (k)) (4) r
mi (σ (k)) ≥ 0, 0 ≤ ai (σ (k)) ≤ 1, 0 ≤ ai (σ (k)) ≤ 1 mi (σ (k)) = 1, ai (σ (k)) + ai (σ (k)) = 1
(5)
i=1
with ai (σ (k)) and ai (σ (k)) being nonlinear weighting functions and mi (σ (k)) regarded as the grades of membership.
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Remark 1: The parameter uncertainties can result in the uncertainties of the membership functions of the plant when we model the plant. In IT2 T-S FM, the uncertain parameter is used to determine the lower and upper membership functions. With the lower and upper membership functions and relevant weighting functions, we can determine the desired membership functions according to the forms of (4) and (8). Thus, the parameter uncertainties can be captured, that is, the uncertain parameter exist in the model in the form of lower and upper bounds. Lam and Seneviratne [35] used an example to demonstrate it specifically. Remark 2: The weighting functions ai (σ (k)) and ai (σ (k)) play an important role in determining the desired membership functions. As the parameter exists in the plant in the form of the uncertainty, the weighting functions should change when the uncertain parameter changes within its range. Thus, they are not necessary to be known. Liang and Mendel [32] chose them as 0.5. In this paper, both are defined as nonlinear functions, which are related to the system state so that they can describe the dynamics of membership functions well. In this paper, the IT2 state-feedback controller does not share membership functions and amount of fuzzy rules with the model for enhancing the flexibility of the controller design. The details of a s-rule controller are as follows. B. IT2 Controller Design Controller Rule j: IF g1 (σ (k)) is Nj1 , and g2 (σ (k)) is Nj2 and, . . . , and g¯ (σ (k)) is Nj¯ , THEN uc (k) = Kj xc (k) , j = 1, 2, . . . , s
(6)
where xc (k) ∈ Rn is the controller state; uc (k) ∈ Rnu is the output of the controller and Kj are controller gains to be determined; s is the number of rules of the controller. The following interval sets describe the firing strength of the jth rule:
j (σ (k)) = ωj (σ (k)), ωj (σ (k)) where ωi (σ (k)) =
¯
uNjq gq (σ (k)) ≥ 0
Fig. 1.
3
Plant prototype.
where ω˘ j (σ (k)) ωj (σ (k)) = s ˘ j (σ (k)) j=1 ω
(8)
ω˘ j (σ (k)) = bj (σ (k))ωj (σ (k)) + bj (σ (k))ωj (σ (k)) 0 ≤ bj (σ (k)) ≤ 1, ωj (σ (k)) ≥ 0, 0 ≤ bj (σ (k)) ≤ 1 s ωj (σ (k)) = 1, bj (σ (k)) + bj (σ (k)) = 1 (9) j=1
bi (σ (k)) and bj (σ (k)) are predefined weighting functions and ωj (σ (k)) is regarded as the grades of membership. Remark 3: In the IT2 controller, the weighting functions bi (σ (k)) and bj (σ (k)) are predefined. Lam et al. [37] have used different values of weighting functions to illustrate whether they have effect on the stability capacity and it has been proven that reasonable values can reduce the conservativeness. To this end, with the above constraints of them, they are also defined as two nonlinear functions related to the system state. Remark 4: For the purpose of reducing the conservativeness, the forms of (4) and (8) are different. If (8) is the same as that of (4), it is obvious that the same form is sjust a specific case of (8) we have designed (when j=1 (bj (σ (k))ωj (σ (k)) + bj (σ (k))ωj (σ (k))) = 1). Therefore, (8) is more general and helpful to enhance the flexibility of controller design.
q=1
ωi (σ (k)) =
¯
C. Communication Links
uNjq gq (σ (k)) ≥ 0
q=1
uNjq gq (σ (k)) ≥ uNjq gq (σ (k)) ≥ 0 ωi (σ (k)) ≥ ωi (σ (k)) ≥ 0 uNjq (gp (σ (k))), uNjq (gp (σ (k))), ωi (σ (k)), and ωi (σ (k)) denote lower membership functions, upper membership functions, the lower grade of membership, and the upper grade of membership, respectively. Then, the overall fuzzy controller is as follows: uc (k) =
s j=1
ωj (σ (k)) Kj xc (k)
(7)
The plant prototype with communication network is shown in Fig. 1. The intermittent data missing phenomenon introduced by network should be considered in this paper. The states of the plant and the controller are not equal [e.g., x(k) = xc (k)], and the output of the controller is not equal to the input of the plant [e.g., uc (k) = u(k)]. A stochastic approach is employed to describe the data missing phenomenon and it is modeled as xc (k) = α (k) x (k) , u (k) = β (k) uc (k) with α(k) and β(k) obeying the Bernoulli process. The processes α(k) and β(k), respectively, describe the imperfect communication link between the sensor and the controller and
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between the controller and the actuator, which are assumed as follows: ¯ α (k) = 1 E {α (k)} = α, Prob {α (k)} = 1 − α¯ α (k) = 0 ¯ β (k) = 1 E {β (k)} = β, . Prob {β (k)} = ¯ 1 − β, β (k) = 0 We introduce the variable e(k) α(k)β(k), then we can get E {e (k)} = e¯ , e (k) = 1 Prob {e (k)} = . 1 − e¯ , e (k) = 0
where ϑ iji1 i2 ...in zl and ϑ iji1 i2 ...in zl are constant scalars to be computed; 0 ≤ υτ is zl (στ (k)) ≤ 1 with = 1 for the property υτ 1zl (στ (k)) + υτ 2zl (στ (k)) τ, s = 1, 2, . . . , n; l = 1, 2, . . . , ς + 1; iτ = 1, 2; ; and υτ σ (k) ∈ Hz is zl (στ (k))
n = 0if otherwise. Thus, q 2 2 2 . . . i1 =1 i2 =1 in =1 τ =1 υτ iτ zl (στ (k)) = 1 for z=1 all l is achieved, which is used to analyze the system stability. Then, (12) can be rewritten as follows: x (k + 1) =
According to the assumption, we have u (k) =
s
ωj (σ (k)) e (k) Kj (k) x (k).
z (k) =
s
i=1
hij (σ (k)) = mi (σ (k)) ωj (σ (k)) =
ωj (σ (k))
j=1 r s
=
mi (σ (k)) ωj (σ (k)) = 1.
z (k) =
i=1 j=1 r s
(11)
mi (σ (k)) ωj (σ (k)) Aij x (k) + Ei w (k) mi (σ (k)) ωj (σ (k)) Cij x (k) + Fi w (k) (12)
where Aij = Ai + e¯ Bi Kj + e˜ (k)Bi Kj , e˜ (k) = e(k) − e¯ Cij = Ci + e¯ Di Kj + e˜ (k)Di Kj .
E {˜e(k)} =
e¯ (1 − e¯ ) , e˜ (k) = e˜ (k)˜e(k) 0, e˜ (k) = e˜ (k).
The following work is to design the controller for the IT2 T-S fuzzy control system (12). The state space H consists of q connected sub-state spaces Hz (z = 1, 2, . . . , q) q with H = ∪z=1 Hz . Then we divide the footprint of uncertainty (FOU) into ς + 1 sub-FOUs. For l = 1, 2, . . . , ς + 1, the expressions of lower and upper membership functions in the lth sub-FOU are as follows: hijl (σ (k)) = hijl (σ (k)) =
q 2 2 z=1 i1 =1 i2 =1 q 2 2 z=1 i1 =1 i2 =1
...
2 n in =1 τ =1
...
n 2 in =1 τ =1
r s
ρijl (σ (k)) (16)
hij (σ (k)) = 1, 0 ≤ ζ ijl (σ (k)) ≤ ζ ijl (σ (k)) ≤ 1 (17)
i=1 j=1
i=1 j=1
It is obvious that
ς+1
× (ζ ijl (σ (k))hijl (σ (k)) + ζ ijl (σ (k))hijl (σ (k)))
According to (3) and (10), the closed-loop system is represented as r s
(15)
l=1
i=1 j=1
x (k + 1) =
hij (σ (k)) Cij x (k) + Fi w (k)
where
From (5) and (9), we know mi (σ (k)) =
i=1 j=1 r s
hij (σ (k)) Aij x (k) + Ei w (k)
i=1 j=1
(10)
j=1
r
r s
ρijl (σ (k)) =
1, 0,
hij (σ (k)) ∈ sub-FOU l else
(18)
where ζ ijl (σ (k)) and ζ ijl (σ (k)) are two functions and unnec-
essary to be known. ζ ijl (σ (k))+ζ ijl (σ (k)) = 1 for i, j and l.
For brevity, hij hij (σ (k)). Before proceeding further, we introduce the following definition. Definition 1 [17]: Under the initial condition x(0) and w(k) ≡ 0, (15) is stochastically stable if there is a matrix W > 0 such that ∞ |x(k)|2 |x(0) < x(0)T Wx(0). E k=0
Our aim in this paper is to design a fuzzy state-feedback controller for nonlinear NCSs with imperfect communication links in the framework of the IT2 T-S FM to satisfy the following two requirements. 1) Equation (15) is stochastically stable. 2) Under zero initial condition, with a given positive scalar γ , the controlled output z(k) satisfies ⎧ ⎫ ∞ ⎨ ⎬ |z (k)|2 ≤ γ w2 . E ⎩ ⎭ k=0
III. M AIN R ESULTS υτ iτ zl (στ (k))ϑ iji1 i2 ...in zl (13) υτ iτ zl (στ (k))ϑ iji1 i2 ...in zl (14)
0 ≤ hijl (σ (k)) ≤ hijl (σ (k)) ≤ 1 ϑ iji1 i2 ...in zl ≤ ϑ iji1 i2 ...in zl
In this section, for given controller gain matrices Kj ( j = 1, 2, . . . , s), we first provide the following theorem, which ensures (15) is stochastically stable with a given H∞ performance. Theorem 1: With the FOU and state space divided into ς + 1 sub-FOUs and q connected sub-state spaces, for the given controller gain matrices Kj ( j = 1, 2, . . . , s) and positive scalar γ , (15) is stochastically stable and satisfies
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a given H∞ performance, if there exist symmetric matrices P > 0, Nijl > 0, Yijl > 0, Xijl > 0, Vijl > 0 (i = 1, 2, . . . , r, j = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1), and M with appropriate dimensions such that the following inequalities hold: r s ¯ 1ijl − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl Wijl ϑ iji1 i2 ...in zl Q 1 2 n i=1 j=1
r s ς+1
ρijl (σ (k)) ζ ijl (σ (k)) + ζ ijl (σ (k))
i=1 j=1 l=1
× hijl (σ (k)) Nijl ≥ 0 −
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))(hijl (σ (k))
i=1 j=1 l=1
(19)
¯ 2ijl + Wijl + M > 0 Q
(20)
−
r s ς+1
¯ 1ijl = ˜ Tij Pˇ ¯ Tij E˜ ijl ˜ ij + ¯ ij − P, ˆ Pˆ = diag P, γ 2 I Q ¯ 2ijl = ˜ ij + ¯ ij − P, ˜ Tij Pˇ 1 ¯ Tij E˜ 1ijl ˆ f = e¯ (1 − e¯ ) Q ¯ Ei ˜ ij = Aij , Pˇ = diag P + Nijl , P + Nijl fBi Kj 0 ¯ Fi Cij ¯ ij = , E˜ ijl = diag I + Xijl , I + Xijl fDi Kj 0 ˇP1 = diag P − Yijl , P − Yijl E˜ 1ijl = diag I − Vijl , I − Vijl A¯ ij = Ai + eBi Kj , C¯ ij = Ci + eDi Kj .
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))(hijl (σ (k)) − hijl (σ (k)) Vijl ≥ 0
(26)
ρijl (σ (k))ζ ijl (σ (k))hijl (σ (k)) Xijl ≥ 0.
(27)
i=1 j=1 l=1
Adding (22)–(25) into (21), it can be found that V (k) ≤ x¯ T (k)
where P > 0 is the matrix to achieve a feasible solution. Let x¯ T (k) = xT (k) wT (k) . Then, one can have the following: V (k) = E {V (k + 1) |¯x (k)} − V (k) T r s Aij PAij ATij PEi ≤ x¯ T (k) x¯ (k) hij ∗ EiT PEi i=1 j=1 P 0 T − x¯ (k) x¯ (k) 0 0 r s ¯ ij − P A¯ Tij PEi ijT P T x¯ (k) = x¯ (k) hij ∗ EiT PEi i=1 j=1
(21)
where
Q1ijl =
ijl
Q2ijl
When w(k) = 0, it can be seen from (19) and (20) that < 0, where =
r s ς+1
r s ς+1
(22)
ρijl (σ (k)) 1 − ζ ijl (σ (k))
˘ 1ij − (hijl (σ (k)) hijl (σ (k)) Q
− hijl (σ (k)))W1ijl + hijl (σ (k)) M1 − M1 + ζ ijl (σ (k)) hijl (σ (k)) − hijl (σ (k)) ˘ 2ij + W1ijl + M1 (29) × Q
W1ijl Wijl = ∗
W2ijl M1 , M= W3ijl ∗
M2 . M3
Then, we can get E xT (k + 1) Px (k + 1) − xT (k) Px (k)
i=1 j=1 l=1
× hijl (σ (k)) − hijl (σ (k)) Wijl ≥ 0
ρijl (σ (k))
i=1 j=1 l=1
in which
#
+ 1 − ζ ijl (σ (k)) × hijl (σ (k)) − 1 M = 0 −
˘ 2ij A¯ T (P − Yijl )Ei Q ij ˘ 2ij = ijT Pˇ 1 ij − P. ,Q = ∗ EiT (P − Yijl )Ei
×
A¯ ij , P¯ = diag{P, P}. fBi Kj
˘ 1ij A¯ T (P + Nijl )Ei Q ij ˘ 1ij = ijT P ˇ ij − P ,Q ∗ EiT (P + Nijl )Ei
We introduce some slack matrices ⎡ r s ς+1 ⎣ ρijl (σ (k)) ζ (σ (k))hijl (σ (k))
ρijl (σ (k))
i=1 j=1 l=1
V(k) = xT (k) Px (k)
i=1 j=1 l=1
r s ς+1
× hijl (σ (k)) × Q1ijl − hijl σ (k) − hijl (σ (k))
× Wijl + hijl (σ (k)) M − M + ζ ijl (σ (k)) hijl (σ (k)) − hijl (σ (k)) (28) × Q2ijl + Wijl + M x¯ (k)
Proof: We firstly prove the stability of (15). Consider the following Lyapunov function for (15):
(25)
i=1 j=1 l=1
where
ij =
(24)
− hijl (σ (k)) Yijl ≥ 0
+ ϑ iji1 i2 ...in zl M − M < 0
where
5
(23)
≤ − λmin (−) xT (k) x (k).
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Computing the sum of the mathematical expectation for both sides of the inequality from k = 0, 1, . . . , d with any d ≥ 1, one can obtain
E xT (d + 1) Px (d + 1) − xT (0) Px (0) d 2 |x (k)| ≤ −λmin (−) E
Adding (26) and (27) to (30), we get J ≤ xT (k)
r s ς+1
+ V(k) +
E
d
|x (k)|
2
$
T
ij ij + ζ ijl (σ (k)))hijl (σ (k))) ∗
k=0
which yields
ρijl (σ (k))(ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
r s ς+1
# C¯ ijT Fi x (k) FiT Fi − γ 2 I
ρijl (σ (k))ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
≤ (λmin (−))−1 xT (0) Px (0)
k=0
− E xT (d + 1) Px (d + 1) ≤ (λmin (−))−1 xT (0) Px (0).
When d = 1, . . . , ∞, considering E{xT (∞)Px(∞)} ≥ 0, we have d 2 |x (k)| ≤ (λmin (−))−1 xT (0) Px (0) E
× zT (k) Xijl z (k) −
ρijl (σ (k))
i=1 j=1 l=1
× ζ ijl (σ (k))(hijl (σ (k)) − hijl (σ (k)))zT (k) Vijl z (k) = x¯ T (k)
r s ς+1
ρijl (σ (k))
i=1 j=1 l=1
×
¯ 1ijl − (hijl (σ (k)) − hijl (σ (k)))Wijl hijl (σ (k)) Q + hijl (σ (k)) M − M ϑ iji1 i2 ...in zl − ϑ iji1 i2 ...in zl x¯ (k)
k=0
= xT (0) Wx (0)
r s ς+1
+ x¯ T (k)
r s ς+1
ρijl (σ (k))ζ ijl (σ (k))
i=1 j=1 l=1
¯ 2ijl + Wijl + M)¯x (k). × (hijl (σ (k)) − hijl (σ (k)))(Q
where W (λmin (−))−1 P
Substituting (13) and (14) into (31), we have
which means W > 0. From Definition 1, the closed-loop fuzzy system in (15) is stochastically stable. The H∞ performance of (15) will be considered. Under the zero initial condition, the H∞ performance index is J E {V (k + 1) |¯x (k)} − V (k) +E zT (k) z (k) |¯x (k) − γ 2 wT (k) w (k) . Then, we get J = E zT (k) z (k) |¯x (k) − γ 2 wT (k) w (k) + V (k) T r s TF Cij Cij Cijl i T hij x (k) ≤ x (k) ∗ FiT Fi i=1 j=1 0 0 − xT (k) x (k) + V(k) 0 γ 2I = xT (k)
+1 r s ς
ρijl (σ (k))(ζ ijl (σ (k))hijl (σ (k))
i=1 j=1 l=1
+ ζ ijl (σ (k)))hijl (σ (k)))
¯ ijT ¯ ij ∗
+ V(k) where
T ¯ ij = C¯ ijT fKjT DTi and V(k) is as in (28).
(31)
C¯ ijT Fi x (k) FiT Fi − γ 2 I (30)
J ≤ x¯ T (k)
r s ς+1 i=1 j=1 l=1
×
2 i2 =1
×
...
2 n
ρijl (σ (k))
q 2 z=1 i1 =1
υτ iτ zl (στ (k))
in =1 τ =1
¯ 1ijl − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl Wijl ϑ iji1 i2 ...in zl Q 1 2 n
+ ϑ iji1 i2 ...in zl M − M + ζ ijl (σ (k)) ϑ iji1 i2 ...in zl − ϑ iji1 i2 ...in zl ¯ 2ijl + Wijl + M ϑ iji i ...i zl × Q 1 2 n −ϑ iji1 i2 ...in zl x¯ (k). (32)
to the proof in [37], at any time, the equality Referring ς+1 ρ (σ (k)) = 1 is ensured by only one ρijl (σ (k)) = 1 ijl l=1 for each fixed i, j value. According to (19) and (20), one can have E zT (k) z (k) − γ 2 wT (k) w (k) + V (k) ≤ 0 % ∞ 2 which yields J ≤ 0. Then, we can obtain E{ k=0 |z(k)| } ≤ γ w2 . The proof is completed. Based on Theorem 1, the existence conditions of the controller can be developed in the following theorem. Theorem 2: Considering the fuzzy system in (15), for a positive scalar γ , (15) is stochastically stable and satisfies a given H∞ performance, if there exist symmetric matrices L > 0, ¯ 1ijl > 0, W ¯ 2ijl > 0, W ¯ 3ijl > 0, N¯ ijl > 0, Xijl > 0, Y¯ ijl > 0, W
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¯ 1, M ¯ 2, M ¯ 3 satisfying the following Vijl > 0 and matrices K¯ j , M conditions: ¯ ijl ˘ ijl 0 (34) ∗ P˘ L − Y¯ ijl < 0 I − Vijl < 0
(35) (36)
where
P˜ ijl = diag N¯ ijl − L, N¯ ijl − L, Xijl − I, Xijl − I P˘ = diag 3L − Y¯ ijl , 3L − Y¯ ijl , 3I − Vijl , 3I − Vijl % ¯ 2ijl ¯ 1ijl ¯ = ϑ iji1 i2 ...in zl , ijl = ¯ 3ijl ∗
˘ ijl =
1ijl EiT
f K¯ jT BTi 2ijl f K¯ jT DTi 0 FiT 0 T T T T ¯ ¯ f Kj Bi 4ijl f Kj Di
˘ ijl = 3ijl ET 0 FiT 0 i ¯ ¯2 ¯ ¯ W2ijl + M −L + W1ijl + M1 Uijl = ¯ 3ijl + M ¯3 ∗ −γ 2 I + W 1ijl = LATi + e¯ K¯ jT BTi , 2ijl = LCiT + e¯ K¯ jT DTi 3ijl = LATi + e¯ K¯ jT BTi , 4ijl = LCiT + e¯ K¯ jT DTi ¯ 1ijl = −ϑ iji1 i2 ...in zl L − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 1ijl 1 2 n ' & 1 ¯1 + ϑ iji1 i2 ...in zl − M rs ¯ 2ijl = − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 2ijl 1 2 n ' & 1 ¯2 + ϑ iji1 i2 ...in zl − M rs ¯ 3ijl = −ϑ iji1 i2 ...in zl γ 2 I − ϑ iji i ...i zl − ϑ iji1 i2 ...in zl W ¯ 3ijl 1 2 n ' & 1 ¯ 3. + ϑ iji1 i2 ...in zl − M rs Furthermore, if the aforementioned conditions hold, the controller gain matrices in (6) can be designed as follows: Kj = K¯ j L−1 . Proof: Firstly, some matrix variables are defined as follows: ¯ 1ijl = LW1ijl L, W ¯ 2ijl = LW2ijl , W ¯ 3ijl = W3ijl L = P−1 , W ¯ 1 = LM1 L, M ¯ 2 = LM2 , M ¯ 3 = M3 K¯ j = Kj L, M N¯ ijl = LNijl L, Y¯ ijl = LYijl L. Pre- and post-multiplying (33) by diag{P1 , P2 } and its transposition, we can obtain ijl ijl
